I meant to warn ya about soldering on those Data East boards, I've messed up a few traces on mine trying to remove components. They have through holes, which means the solder will attach itself to both the top and bottom of the board. Recommend using a dab of flux and solder wick or a heated solder sucker. But you don't want to apply toooo much heat as the traces on these board seem very delicate, more so than any other I've messed with.
If ya start to get frustrated..... back away from it. (trust me)

That red component is a resistor pack, never seen one of those burn like that, especially for control inputs. I would continue tracing and find out where it leads to from that resistor pack trace.
My Captain America inputs seem to lead back to a fancy surface mount chip, hopefully yours doesn't. I don't have any other Data East boards to look at and compare to yours.
Still really surprised that only ONE input has been effected on yours.
Wish the schematics in the manual for yours were a bit better.....drawn the way they are makes it harder to figure out without printing the whole dang thing off to bounce from page to page. (for me anyways)
http://arcarc.xmission.com/PDF_Arcade_Manuals_and_Schematics/Heavy%20Barrel.pdf